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PSY 512: Advanced Statistics for

Psychological and Behavioral Research 2

�What is SEM?

�When should we use SEM?

�What can SEM tell us?

�SEM Terminology and Jargon

�Technical Issues

�Types of SEM Models

�Limitations of SEM

� Combination of path analysis and factor analysis

• Path analysis: concerned with the predictive ordering of

measured variables

� X �Y � Z

• Factor analysis: concerned with latent factors (i.e.,

unmeasured variables)

� Latent construct of “intelligence”

� SEM is also known as:

• Causal modeling

• Covariance structure analysis

• Latent variable modeling

• “LISREL” modeling

• Simultaneous equations

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�Psychologists focus a great deal of attention on latent variables• We are unable to measure these directly so we must rely

on measurable indicators

�An operationalization of data as an abstract construct • A data reduction method that uses “regression like”

equations

• Takes many variables and attempts to explain them with a one or more “factors”

• Correlated variables are grouped together and separated from other variables with little or no correlation

Latent

Variable

Variable 1 Variable 2 Variable 3 Variable 4

Intelligence

Raven’s

Matrices

Miller

Analogies

Wechsler

Adult

Intelligence

Scale

Wonderlic

Test

Intelligence is a “factor”

because it is an

unmeasured variable that is

hypothesized to cause the

covariation among a set of

measured variables

The measured variables

(e.g., Raven’s Matrices)

are referred to as

“indicators”

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Latent Variable (Factors, Constructs)

Measured Variable (Observed Variable, Indicator Variable, Manifest Variable)

Regression Weight or Factor Loading

Covariance

� Endogenous Factor: A factor that the model attempts to explain

(i.e., has one or more straight lines with arrows pointed at it)

� Exogenous Factor: A factor that the model does not attempt to

explain (i.e., no straight lines with arrows pointed at it)

� Induced Variable: An unmeasured variable (like a factor) that is

believed to be the result of other variables in the model

• Example: Stress (latent variable) may be determined by features such as role

ambiguity, negative life events, and social conflict

� Full Structural Equation Model: Combines a measurement model

and a structural model (includes associations between unmeasured

and measured variables)

• Measurement Model: Relates unmeasured variables to measured variables

(similar to factor analysis)

• Structural Model: Summarizes relationships between unmeasured variables

(similar to path analysis)

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Locus of Control,

Self-Esteem,

Overall

Satisfaction, and

Loneliness are

factors

Locus of Control

and Loneliness

are exogenous

factors…Self-

Esteem and

Overall

Satisfaction are

endogenous

factors

Planunhy,

planwork, and

planbett are

indicators of

Locus of Control

Each of the other

factors has its

own set of

indicators

The empty ovals

represent various

types of error

that influence the

indicators and

the endogenous

factors

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The four straight

lines connecting

the factors

represent

hypothesized

direct effects of

one factor on

another

The curved two-

headed arrow

connecting

Loneliness and

Locus of Control

indicates a

relationship

between those

factors

� η (eta): Dependent (endogenous) variable that is unmeasured

(unobserved, latent)

� y: Indicator of dependent variable that is measured

(observed, manifest)

� ξ (xi or ksi): Independent (exogenous) variable that is

unmeasured (unobserved, latent)

� x: Indicator of independent variable that is measured

(observed, manifest)

� ε (epsilon): Error in observed dependent variable

� δ (delta): Error in observed independent variable

� ζ (zeta): Sources of variance in η not included among the ξs

(disturbances, error in equations, unexplained error in model)

� λy (lambda): Coefficients relating unmeasured dependent variables

to measured dependent variables

� λx (lambda): Coefficients relating unmeasured independent

variables to measured independent variables

� β (beta): Coefficients interrelating unmeasured dependent variables

� γ (gamma): Coefficients relating unmeasured independent variables

to unmeasured dependent variables

� φ (phi): Variances and covariances among unmeasured independent

variables

� ψ (psi): Variances and covariances among disturbances

� Θε (theta): Variances and covariances among errors in measured

dependent variables

� Θδ (theta): Variances and covariances among errors in measured

independent variables

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�The variables should be measured at either the interval or ratio scale

�The variables should have multivariate normal distributions

�The model should be correctly specified (e.g., relevant variables should be included and the direction of causal flow should be correct)

�SEM requires large samples for reliable results• More participants are needed for more complex

models, models with weak relationships, models with few measured variables, and models with variables that have nonnormal distributions

�The first step in modeling is the specification of a model• This should be based as much as possible on

previous knowledge

• If theory suggests competing models, then they should each be specified

�After a model is specified, the next step is to obtain parameter estimates (i.e., estimates of the coefficients representing direct effects, variances, and covariances)• This is accomplished using SEM software such as

AMOS

�AMOS will determine the estimates that will

most nearly reproduce the matrix of

observed relationships (i.e., correlation

matrix)

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The SEM program will determine the estimates that most nearly reproduce the

matrix of observed correlations. For example, Lifesat and Planunhp have a

correlation of -.12. The SEM program will look at the various ways of connecting

those two variables in your model and try to reproduce the observed correlation

of -.12 (while doing the same thing for every other observed association)

The numbers beside

the lines represent the

magnitude of the

effects. The numbers

at the tails of the

arrows represent the

variance of errors. In

this model, Self-

Esteem has a positive

association with

Overall Satisfaction

whereas Loneliness

has a negative

association with

Overall Satisfaction

� Evaluating SEM involves the following…

• Theoretical Criteria

� The model parameters should be assessed from a theoretical perspective (e.g.,

are the signs and magnitudes of the coefficients consistent with what is known

from previous research?)

• Statistical Criteria

� Identification status of the model (i.e., is there a unique solution for each

parameter in the model?)

� The reasonableness of the parameters (e.g., negative variances or correlations

greater than 1 will alert you to problems)

• Assessment of Fit

� The goal of SEM is to produce estimates that most nearly reproduce the

relationships in the original correlation matrix

� Fit indices determine how well the model matches the observed relationships

� Researchers want fit indices that indicate a good fit (e.g., statistics that say any

differences that appear could have occurred by chance)

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� Comparative Fit Index (CFI): Compares the proposed

model to an independence model (where nothing is

related)

� Root Mean Square Error of Approximation (RMSEA):

Compares the estimated model to a saturated or perfect

model

� Other popular fit indices• Normed Fit Index (NFI)

• Goodness of Fit Index (GFI)

• Incremental Fit Index (IFI)

• Non-normed Fit Index (NNFI)

• Akaike Information Criterion (AIC)

• Bayesian Information Criterion (BIC)

• Chi-square

�After examining the results of your SEM model,

you may decide to modify the model

• Many SEM programs will provide suggestions about

alterations to the model

�These modifications are “data-driven” and

there is the risk of capitalizing on chance (e.g.,

fitting the model to the peculiarities of one

particular sample)

• The probability values (p values) associated with a

model including data-driven modifications are not

accurate (to an unknown extent)

�We have discussed full structural equation

models so far…but there are other models

that are useful

• Simple submodels

� Confirmatory factor analysis (CFA)

� Path analysis

• Advanced extensions

� Longitudinal models

� Multisample models

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�CFA involves only a measurement model• Only models the direct effects of the factors on the

measured variables, the covariances among the factors,

and the errors of measurement

• Does not include specification of a structural model that

relates the factors to each other

�Unlike Exploratory Factor Analysis (EFA), CFA

allows the researcher to specify that particular

factors affect (or load on) particular measured

variables whereas all factors affect all

measured variables in EFA

�Path analysis only involves measured

variables

• No latent variables are involved

�There is a structural model without a

traditional measurement model (i.e., all of

the coefficients are fixed at 1)

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�SEM is very useful for the analysis of

longitudinal data such as panel data (i.e.,

measurements for the same individuals at

two or more time points)

• Allows us to develop a clearer idea about temporal

sequencing than can be drawn from a single point in

time

• With three or more time points it is possible to

estimate models with cross-lagged and synchronous

effects

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�An SEM model can be fit to two or more

groups simultaneously which allows for an

assessment of difference in the degree of fit

between the groups

• Example: The same model linking intelligence and

mental rotation abilities could be fit for men and

women simultaneously. This would allow researchers

to see if the model fit better for one sex than the

other

�Recursive

• Direction of influence one direction

� No reciprocal causation

� No feedback loops

• Disturbances not correlated

�Nonrecursive

• Either reciprocal causation, feedback loops, or

correlated disturbances

y2x1 y3

y3

y2

x3

x1

y1

x2

Model 1

Model 2

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x2 y1

x1 y2

y3

y2

x3

x1y1

x2

Model 1

Model 2

� SEM is a confirmatory approach

• Researchers need to have hypotheses about the relationships

• SEM is not “causal modeling”

� SEM is “correlational” but it can be used with experimental data

• Mediation and manipulation can be tested

� SEM is a sophisticated technique but it does not make up for

poor research design

� Biggest limitation is sample size

• It needs to be large to get stable estimates of the covariances/correlations

• It generally needs at least 200 participants for even small models

• A minimum of 10 participants are required for each estimated parameter

� SEM has proven to be a very versatile tool

� One strength of SEM is the requirement of prior knowledge

of the phenomena under examination

• In practice, this means that the researcher is testing a model that is

based on an exact and explicit plan or design

• Very complex and multidimensional structures can be measured

with SEM

• SEM is the only linear analysis method that allows complete and

simultaneous tests of all relationships

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� Limitations of SEM• Researchers must be very careful with the study design when using

SEM for exploratory work (i.e., SEM does not equal “causal

modeling”)

• SEM is complex and it is often done poorly

� A lot of jargon without a clear understanding

• Overgeneralization is always a problem and this is certainly true for

SEM

• SEM is based on covariances which are only stable when estimated

from large samples (i.e., at least 200 observations)

� …but really large samples can cause problems for some of the fit

statistics (e.g., leads to significant χ2)

• SEM programs allow calculation of modification indices which help

researcher to fit the model to the data

� ...but overfitting models to the data reduces generalizability!

�1. Work through the example in Amos • This can be done by following the instruction in the

Amos Tutorial (pp. 1-21)

• Print the path diagram and bring it to class

�2. Fit another model using data from another source• This can be data from your lab, found on the internet

(or other location), or fabricated

• The important thing is that you work through the process of SEM

• Print the path diagram, bring it to class, and be prepared to discuss what you did